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$\textbf{Context:}$ While just browsing the site, I came across this integral in this post:

$$\int_0^{10}x^2\mathrm d\left\{x+\frac12\right\}$$

$\textbf{Doubt 1:}$ $\dfrac{\mathrm d\left\{x+\frac12\right\}}{\mathrm dx}$ is not defined for half-integral values of $x$. Then, how does the above integral make sense as this differential is not defined at $x=\frac12,\frac32\cdots\frac{19}2$?!

$\textbf{My attempt:}$ Let's assume for a moment that this notation makes sense. Since $\left\{x+\frac12\right\}$ is not differentiable when $x$ is a half-integer, we need to split the integral such that the differential is defined for the values of $x$ lying in the open interval described by the bounds of each integrral: $$\int_0^{10}x^2\mathrm d\left\{x+\frac12\right\}=\int_0^\frac12x^2\mathrm dx+\int_\frac12^\frac32x^2\mathrm dx\cdots\int_\frac{19}2^{10}x^2\mathrm dx$$

Now, using the below property repeatedly

$$\displaystyle\int_a^bf(x)\mathrm dx+\displaystyle\int_b^cf(x)\mathrm dx=\displaystyle\int_a^cf(x)\mathrm dx$$

we can rewrite the integral as $\displaystyle\int_0^{10}x^2\mathrm dx$. But, this is wrong as per this answer. It also states that this is a Riemann-Stieltjes integral which I don't understand at all(I'm in high school).

$\textbf{Doubt 2:}$ Where did I make a mistake in simplifying the given integral?

Please help me out with my doubts. I'm not able to find what is wrong with my procedure and am really perplexed.

Tijil Saka
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Integreek
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    Unfortunately, you have to study Riemann -Stieltje's integration for this. This is not a Riemann integral. – Kavi Rama Murthy Feb 07 '25 at 09:57
  • @geetha290krm Is it possible for me to understand what Riemann integrals are? I tried reading the Wikipedia article on this, but everything after the Overview section went over my head. – Integreek Feb 07 '25 at 10:10
  • "Is it possible for me to understand what Riemann integrals are?" Yes, but it'll take time. Riemann integral is not a formula that you read and understand, it's a whole theory on integration (it's the first rigorous formulation actually). – Sine of the Time Feb 07 '25 at 10:24
  • @SineoftheTime in which year of college is this taught? – Integreek Feb 07 '25 at 10:39
  • I don't know; I guess usually in the first year. You can find it in any real analysis book. – Sine of the Time Feb 07 '25 at 11:07
  • It depends. In the US where I live, real analysis is typically taught to university students around their junior/senior year if they follow their "road map". But the year a student learns the subject varies depending on many factors, such as: how many AP/IB or dual enrollment classes a person took; when a person switches their career path; or if the person succeeds in the prerequisite classes. Regardless of when people learn it, you could begin learning real analysis by starting on Page 1 of any standard real analysis textbook assuming you understand the basics of proofs, functions, and logic. – Accelerator Feb 07 '25 at 11:12
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    You can avoid Riemann Stieltje's integration by using integration by parts with u=x^2, v={x+1/2}. – homo_morph Feb 09 '25 at 04:26
  • @homo_morph thanks, but what about this question? IBP still gives a Riemann Stieltjes integral in this case. – Integreek Feb 09 '25 at 05:09
  • you can do this by negative dirac delta but its also not taught in high school – Blue Cat Blues Mar 24 '25 at 18:26
  • @MathStackexchangeIsMarvellous bhaiya, ye cheez konse college year mein sikhayi jaati hai? – Integreek Mar 25 '25 at 05:31
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    idk...ho sakta hai na bhi sikhayi jaaye...agar engg karoge to usme in sab ki uthni zaroorat nahi padti hai (unless math specific branch na ho)...i didnt study this in clg (upar upar se aata hai mujhe ye) – Blue Cat Blues Mar 25 '25 at 07:00
  • @MathStackexchangeIsMarvellous and, I wanted to ask why did you delete your answer to my question of $\int\frac{\mathrm dx}{a\sin x+b\cos x+c}?$ – Integreek Mar 26 '25 at 18:18
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    @Integreek beacuse it was incorrect....i used a theory of definite integrals and applied it to indefinite integral – Blue Cat Blues Mar 26 '25 at 18:32
  • @MathStackexchangeIsMarvellous but definite integrals can be related to indefinite integrals by the FTC, so maybe the method you used could indeed be applied to my question? And sorry to go off-topic bhaiya, but I think should update your profile. – Integreek Mar 27 '25 at 05:47
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    @Integreek no no...the person who corrected me was right...and yes they can be related by FTC but I used complex analysis to evaluate that (and that approach was incorrect)...btw the best method was to convert asinx+bcosx into Rain(x+theta) – Blue Cat Blues Mar 27 '25 at 06:49
  • And what should I update in my profile? @Integreek – Blue Cat Blues Mar 27 '25 at 06:50
  • @MathStackexchangeIsMarvellous your age, it says 18 year old fresher. – Integreek Mar 27 '25 at 15:24
  • @Integreek im in first year...and ill turn 19 in august – Blue Cat Blues Mar 27 '25 at 20:40
  • @MathStackexchangeIsMarvellous oh..really sorry for that….actually on the internet, it was showing that complex analysis is taught in second or third year; I shouldn’t have assumed…it’s nice then, isi saal complex analysis seekhne ko milega! – Integreek Mar 28 '25 at 09:18
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    @Integreek complex analysis was taught to me in my first semester in my college...best of luck (complex analysis is exhilarating) – Blue Cat Blues Mar 28 '25 at 11:16

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